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Theorem carden2b 8391
Description: If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 8390 are meant to replace carden 8965 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
carden2b  |-  ( A 
~~  B  ->  ( card `  A )  =  ( card `  B
) )

Proof of Theorem carden2b
StepHypRef Expression
1 cardne 8389 . . . . 5  |-  ( (
card `  B )  e.  ( card `  A
)  ->  -.  ( card `  B )  ~~  A )
2 ennum 8371 . . . . . . . 8  |-  ( A 
~~  B  ->  ( A  e.  dom  card  <->  B  e.  dom  card ) )
32biimpa 486 . . . . . . 7  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  B  e.  dom  card )
4 cardid2 8377 . . . . . . 7  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
53, 4syl 17 . . . . . 6  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  ( card `  B
)  ~~  B )
6 ensym 7616 . . . . . . 7  |-  ( A 
~~  B  ->  B  ~~  A )
76adantr 466 . . . . . 6  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  B  ~~  A )
8 entr 7619 . . . . . 6  |-  ( ( ( card `  B
)  ~~  B  /\  B  ~~  A )  -> 
( card `  B )  ~~  A )
95, 7, 8syl2anc 665 . . . . 5  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  ( card `  B
)  ~~  A )
101, 9nsyl3 122 . . . 4  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  -.  ( card `  B
)  e.  ( card `  A ) )
11 cardon 8368 . . . . 5  |-  ( card `  A )  e.  On
12 cardon 8368 . . . . 5  |-  ( card `  B )  e.  On
13 ontri1 5467 . . . . 5  |-  ( ( ( card `  A
)  e.  On  /\  ( card `  B )  e.  On )  ->  (
( card `  A )  C_  ( card `  B
)  <->  -.  ( card `  B )  e.  (
card `  A )
) )
1411, 12, 13mp2an 676 . . . 4  |-  ( (
card `  A )  C_  ( card `  B
)  <->  -.  ( card `  B )  e.  (
card `  A )
)
1510, 14sylibr 215 . . 3  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  ( card `  A
)  C_  ( card `  B ) )
16 cardne 8389 . . . . 5  |-  ( (
card `  A )  e.  ( card `  B
)  ->  -.  ( card `  A )  ~~  B )
17 cardid2 8377 . . . . . 6  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
18 id 23 . . . . . 6  |-  ( A 
~~  B  ->  A  ~~  B )
19 entr 7619 . . . . . 6  |-  ( ( ( card `  A
)  ~~  A  /\  A  ~~  B )  -> 
( card `  A )  ~~  B )
2017, 18, 19syl2anr 480 . . . . 5  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  ( card `  A
)  ~~  B )
2116, 20nsyl3 122 . . . 4  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  -.  ( card `  A
)  e.  ( card `  B ) )
22 ontri1 5467 . . . . 5  |-  ( ( ( card `  B
)  e.  On  /\  ( card `  A )  e.  On )  ->  (
( card `  B )  C_  ( card `  A
)  <->  -.  ( card `  A )  e.  (
card `  B )
) )
2312, 11, 22mp2an 676 . . . 4  |-  ( (
card `  B )  C_  ( card `  A
)  <->  -.  ( card `  A )  e.  (
card `  B )
)
2421, 23sylibr 215 . . 3  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  ( card `  B
)  C_  ( card `  A ) )
2515, 24eqssd 3478 . 2  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  ( card `  A
)  =  ( card `  B ) )
26 ndmfv 5896 . . . 4  |-  ( -.  A  e.  dom  card  -> 
( card `  A )  =  (/) )
2726adantl 467 . . 3  |-  ( ( A  ~~  B  /\  -.  A  e.  dom  card )  ->  ( card `  A )  =  (/) )
282notbid 295 . . . . 5  |-  ( A 
~~  B  ->  ( -.  A  e.  dom  card  <->  -.  B  e.  dom  card ) )
2928biimpa 486 . . . 4  |-  ( ( A  ~~  B  /\  -.  A  e.  dom  card )  ->  -.  B  e.  dom  card )
30 ndmfv 5896 . . . 4  |-  ( -.  B  e.  dom  card  -> 
( card `  B )  =  (/) )
3129, 30syl 17 . . 3  |-  ( ( A  ~~  B  /\  -.  A  e.  dom  card )  ->  ( card `  B )  =  (/) )
3227, 31eqtr4d 2464 . 2  |-  ( ( A  ~~  B  /\  -.  A  e.  dom  card )  ->  ( card `  A )  =  (
card `  B )
)
3325, 32pm2.61dan 798 1  |-  ( A 
~~  B  ->  ( card `  A )  =  ( card `  B
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1867    C_ wss 3433   (/)c0 3758   class class class wbr 4417   dom cdm 4845   Oncon0 5433   ` cfv 5592    ~~ cen 7565   cardccrd 8359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-ord 5436  df-on 5437  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-er 7362  df-en 7569  df-card 8363
This theorem is referenced by:  card1  8392  carddom2  8401  cardennn  8407  cardsucinf  8408  pm54.43lem  8423  nnacda  8620  ficardun  8621  ackbij1lem5  8643  ackbij1lem8  8646  ackbij1lem9  8647  ackbij2lem2  8659  carden  8965  r1tskina  9196  cardfz  12169
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